Abstract Algebra: Theory and Applications · Problem 15.8.5
Rings — Thomas Judson
Mathematics — Abstract Algebra · sylow-theorems
Problem
Example 15.2.11 shows that every group of order 48 has a normal subgroup. The dicyclic groups are an infinite family of non-abelian groups with order 4n, which includes the quaternions (the case of n=2). So the permutation group DiCyclicGroup(12) has order 48. Use Sage to follow the logic of the proof in Example 15.2.11 and construct a normal subgroup in this group. (In other words, do not just ask for a list of the normal subgroups from Sage, but instead trace through the implications in the example to arrive at a normal subgroup, and then check your answer.)